Buckner gives an argument, nicely laid out in numbered steps. (3), if
I remember them right, is that children can understand the concept of two
(twoness). (4), supposedly following from this, is that they can
understand the concept of number (which in turn contributes to an argument
that the concept of number does not depend on "Hume's Principle"). I'm not
sure about this step. Maybe they can understand ONE and TWO and THREE
without having a clear conception of what the determinable is of which
these are determinates?
I haven't looked at Piaget in years, but isn't therre a stage at which
children can count a bunch of objects, then watch them being rearranged,
agree that none were added or taken away, but still have to count them
again before realizing that the number is still the same? This would make
sense if their understanding of the specific numbers was tied to a
particular geometrical gestalt ("FIVE is when the things are aranged in a
quincunx, or can obviously be rearranged into one"), and one could have
concepts like this of several small numbers without mastering the general
concept of CARDINAL NUMBER.
(Oh. May I sit out the quarrels about Frege and the relevance or
irrelevance of psychology to logic? Even if the Fregean analysis reveals a
nice logical structure, maybe it isn't the only interesting structure in
the neighborhood, and surely even the most puritanical Fregean could admit
that it is-- at least in principle-- possible for psychological data to
suggest or render salient some logical structure!)
--
Allen Hazen
Philosophy Department
University of Melbourne
interests: lots of things in logic & philosophy of math